Ms McVey, who quit as work and pensions secretary last November in protest at Mrs May’s deal, has already announced she will stand in the forthcoming leadership contest.

Launching the Blue Collar Conservatism group at an event in Westminster, she said the next leader of the party must be a “Brexiteer who believes in Brexit”.

‘Broken trust’

She added that the Tories’ “failure to deliver Brexit” meant the party had failed to capitalise on winning over voters who abandoned Labour at this month’s local elections.

“A majority of these voters voted to leave the EU, and on this we have broken their trust,” she said.

The Leave-supporting MP added it was “essential” for the UK to leave the EU before the new deadline of 31 October, with or without a deal.

She also called for international aid spending to be lowered to 2010 levels, which she said would free up £7bn in extra funding for schools and the police.

Doing so would allow the Conservatives to “match people’s needs and priorities”, she added.

According to the Mail on Sunday, Ms McVey will embark on a “pub tour” campaign, where she will also call for the HS2 rail project to be scrapped.

Meanwhile, the One Nation Conservative Caucus is later expected to publish a list of demands for whoever wins the Tory leadership race.

The group is also expected to call for further action on climate change, and reportedly aims to block any candidate who backs a no-deal Brexit.

Analysis by Ben Wright, BBC political correspondent

The unofficial contest to replace Theresa May is already up and running, with candidates declaring their desire for the job and policy platforms being set out.

Work and Pensions Secretary Amber Rudd has not – yet – thrown her hat in the ring but spearheads a 60-strong block of Tories called the One Nation Caucus.

On Monday, it will launch a declaration of values that seeks to cement the post-Theresa May Tory Party in the centre ground.

Sources deny it’s a “stop Boris Johnson” effort, but the group is emphatic there must be no future coalition between the Conservatives and The Brexit Party and is stressing the importance of issues like the environment.

Amber Rudd, who spoke at an event to launch the group, has said it is “entirely possible” she will launch a bid for the leadership once Mrs May steps down.

Speaking at the launch she said: “The Conservative Party is entering a new phase and we here in this room are determined to shape that phase.”

Former Brexit Secretary Dominic Raab, Health Secretary Matt Hancock and Chief Secretary to the Treasury Liz Truss will speak also at a Telegraph event on the future of the Conservative Party later this evening.

All three have been tipped as contenders.

In a Conservative leadership contest, MPs hold a series of ballots, with the candidate gaining the fewest votes eliminated at each stage.

Once the field is reduced to two, the winner is chosen by a vote of party members.